Fractals/island wake

The biggest island of the p/q wake for p/q <= 1/3 is located on the first branch ( reading from the left to right) has period = q+1 external rays ? for p/q > 1/3 is located on the last branch ( reading from the left to right) has period = q+2 Examples Wake 1/2 period 3 island is the biggest island of wake 1/2 find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7) Plane description :R2F(1/2B1)S by Robert P. Munafo, 2008 Feb 28. -1.76733 +0.00002 i @ 0.05 One can check it using program Mandel by Wolf Jung : The angle 3/7 or p011 has preperiod = 0 and period = 3. The conjugate angle is 4/7 or p100 . The kneading sequence is AB* and the internal address is 1-2-3 . The corresponding parameter rays are landing at the root of a primitive component of period 3. See also: family: real slice of Mandelbrot set. periodic part: period doubling cascade. Escape route 1/2 the Myrberg-Feigenbaum point of family chaotic part main antenna is a shrub of family 1/3 Period 4 island Center X -0.15710375803 Center Y +1.03258348530 Pixel step +0.0000375 wake 1/7 12/25 wake 12/25 root c = -0.738203140939397 +0.124839088573366 i The 12/25-wake of the main cardioid is bounded by the parameter rays with the angles 11184809/33554431 or p0101010101010101010101001 and 11184810/33554431 or p0101010101010101010101010 . the center of the satellite component c = -0.739829393511579 +0.125072144080321 i period = 25 biggest island of 12/25 wake cardioid center c = -0.744245042107463 +0.127908444364520 i period = 27 cusp Size miniset-and-embedded-julia-size-estimates Windows of periodicity scaling by E Demidov J.A.Yorke, C.Grebogi, E.Ott, and L.Tedeschini-Lalli "Scaling Behavior of Windows in Dissipative Dynamical Systems" Phys.Rev.Lett. 54, 1095 (1985) B.R.Hunt, E.Ott Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map J.Phys.A 30 (1997), 7067. Haskell program size formula degree p a b = do -- z = x + i y = 0 x <- real y <- real x .= 0 y .= 0 -- matrix L lxa <- real lxb <- real lya <- real lyb <- real -- L = identity lxa .= 1 lxb .= 0 lya .= 0 lyb .= 1 -- matrix B bxa <- real bxb <- real bya <- real byb <- real -- B = identity bxa .= 1 bxb .= 0 bya .= 0 byb .= 1 -- loop one period j <- int j .= 1 while_ (j .< p) $ do -- allocate next z, l xn <- real yn <- real lxan <- real lxbn <- real lyan <- real lybn <- real -- calculate next z, l let R2 fx fy = formula (R2 a b) (R2 x y) -- calculate derivatives fdxa = ddx d x fx fdxb = ddx d y fx fdya = ddx d x fy fdyb = ddx d y fy d v u | u == x && v == x = lxa | u == x && v == y = lxb | u == y && v == x = lya | u == y && v == y = lyb | u == v = 1 | otherwise = 0 -- z = f(z, c) xn .= fx yn .= fy x .= xn y .= yn -- L = J_f(z, c) lxan .= fdxa lxbn .= fdxb lyan .= fdya lybn .= fdyb lxa .= lxan lxb .= lxbn lya .= lyan lyb .= lybn -- B = B + 1/L det <- real det .= lxa * lyb - lxb * lya bxa .= bxa + lyb / det bxb .= bxb - lxb / det bya .= bya - lya / det byb .= byb + lxa / det -- loop counter j .= j + 1 -- l = sqrt (abs (det L)) l <- real l .= sqrt (abs (lxa * lyb - lxb * lya)) -- beta = sqrt (abs (det B)) beta <- real beta .= sqrt (abs (bxa * byb - bxb * bya)) -- compute l^d b d <- float d .= degree / (degree - 1) llb <- real llb .= l ** d * beta return_ (1 / llb) References